Many people find the graphic approach more interesting because it gives a visual context for factors. It also helps make connections between factor pairs, such as 2 and 12, and 4 and 6. It may encourage systematic thinking and a means of discussing how all the possible factors were found. It also makes the pattern approach explained in Teacher B's method a little easier to follow. And, as you've seen, it can provide a means to extend the generalization, since it is a way to represent the structure of what a factor is.

Question: What is similar or different about Teacher A's and Teacher B's reasoning? Compare the two.

Both answers show how to systematically check all possible factors. Teacher A makes and tests a conjecture regarding the lack of new, undiscovered factors after 8. Although Teacher A's example is successful because it shows systematic testing of all possible values of factors, it would be a tedious method to follow for larger numbers. Furthermore, it is not always possible to prove something by testing all possible values, nor is this an adequate method of proof.

Teacher B has a more sophisticated way of checking and explaining, and shows emerging understanding of factoring expressions, such as 2 x 12, to think "2 x 2 x 6" and then to write "4 x 6." Her method draws on connections to the rectangle representations to use reasoning about rearranging the area to give new factors. Her method also starts a general observation about the relationship between factors but relies on a specific case, that of an 8-by-3 rectangle.

Question: What type of reasoning does Teacher C use in explaining that there isn't going to be a new factor of 24 after 5?

Teacher C is systematically thinking about possibilities for factors without testing each one. Teacher C's method has sophisticated reasoning that relies on a logical argument: First he assumes that there is another factor that hadn't been found. Then he shows that such a factor cannot exist because larger factors are paired with smaller factors of a given product, and all the smaller factors are accounted for. This is the kind of reasoning that is central to proof by contradiction in mathematics –– assuming that a solution exists that is not on the established list of solutions, and then reasoning that this solution must be impossible.